Additional Properties of the Binomial Distribution:
*** View copy of Table 4, Appendix II. (This will be valuable to you!)
The graph of the binomial distribution is in the from of a histogram. The values of r or p
(successes) are placed along the horizontal axis and the values of P(r) on the vertical axis.
The binomial distribution is a discrete probability distribution because r can assume only
whole number values such as 0, 1, 2, 3, and so on. Each bar must be only 1 unit wide
because the sum of the areas of the bars must be 1. There is always a break symbol on
your horizontal axis.
*** View example #5 text p. 281.
Whenever p equals .5, the graph of the binomial distribution will be symmetrical no
matter how many trials we have.
Mean and Standard Deviation of Binomial Probability Distributions
Two features that help describe the graph of any distribution are the balance point
of the distribution and the spread of the distribution about the balance point.
Balance Point – mean (μ) of the distribution.
The mean is the expected value of the number of successes.
Measure of Spread – most commonly used.
- one is the standard deviation (σ).
For the binomial distribution there are 2 formulas we can use to compute the mean (μ)
and the standard deviation (σ).
For the Binomial Distribution:
μ = np
σ = √npq
where n is the number of trials, p is the probability of success, and q is the probability of
failure (q = 1 – p).
*** View example 6 (text p. 284).
*** View guided exercise 7 (text p. 284 - 295).
The mean μ is not only the balance point of the distribution, it is also the expected value
of r.
Examples:
1) The probability that a restaurant patron will request seating in the nonsmoking section
is 0.65. A random sample of 5 people call to make reservations. Let r be the number
who request seating in the nonsmoking section.
a) Find P(r) for r = 0, 1, 2, 3, 4, 5.
b) Make a histogram of the r probability distribution?
c) What is the expected number of people out of the 5 who will request a
nonsmoking section?
d) What is the standard deviation of the r probability distribution?
2) Long-term history has shown that 65% of all elected offices in the Gunnison County,
Colorado have been won by Republican candidates. This year there are 5 public offices
up for election. Use the above estimate for the probability of a Republican being elected
to office and let r be the number of public offices won by Republicans.
a) Find P(r) for r = 0, 1, 2, 3, 4, 5.
b) Make a histogram for the r probability distribution.
c) What is the expected number of Republicans who will win office in the coming
election?
d) What is the standard deviation of the r distribution?
e) Find the probability that three or more Republicans will win office.
f) Find the probability that two or fewer Republicans will win office.
3) A student has an option of using the American Heritage Dictionary, Webster’s, or
Random House in English 103. About 1/3 of the students use Webster’s. 2/3 of the
students use one of the other dictionaries.
a) Find the probability that out of 4 English 103 students 3 or more use Webster’s.
b) If 275 students are registered for English 103 next term and each student buys a
dictionary at the bookstore, what is the expected number of Webster’s Dictionaries the
bookstore will need?
4) Jim is an automobile salesman at Courtesy Cars, Inc. He has a history of making a
sale for about 15% of all the customers to whom he shows automobiles and goes out on
test drives with eight customers. Use the above estimate for the probability of a single
sale and let r be the number of sales on a day when Jim has eight customers.
a) Find P (r) for r = 0, 1, 2, 3, 4, 5, 6, 7, 8.
b) Make a histogram for the r probability distribution.
c) What is the expected number of cars Jim will sell if he has eight customers?
d) What is the standard deviation of the r distribution?
e) What is the probability that Jim will sell at least one car on a day when he has
eight customers?
f) What is the probability that Jim will sell two or fewer cars on a day when he
has eight customers?